Scott Aaronson (MIT) Based on joint work with John Watrous (U. Waterloo)

1. Quantum Computing With Closed Timelike Curves BQP Scott Aaronson (MIT) Based on joint work with John Watrous (U. Waterloo) PSPACE

2. Motivation Ordinary quantum computing too pedestrian In the past, CTCs have mostly been studied from the perspective of GR Studying them from a computer science perspective leads us to ask new questions—like, how hard would Nature have to “work” to ensure causal consistency? Hopefully, leads to some new insights about causality, linearity of quantum mechanics, space vs. time, ontic vs. epistemic…

3. Bestiary of Complexity Classes EXP PSPACE The difference between space and time in computer science: you can reuse space, but not time BQP P

4. Everyone’s first idea for a CTC computer: Do an arbitrarily long computation, then send the answer back in time to before you started This does not work. • Why not? • Ignores the Grandfather Paradox • Doesn’t take into account the computation you’ll have to do after getting the answer

5. Deutsch’s Model A closed timelike curve (CTC) is simply a resource that, given an operation f:{0,1}n{0,1}n acting in some region of spacetime, finds a fixed point of f—that is, an x s.t. f(x)=x Of course, not every f has a fixed point—that’s the Grandfather Paradox! But since every Markov chain has a stationary distribution, there’s always a distribution D such that f(D)=D Probabilistic Resolution of the Grandfather Paradox- You’re born with ½ probability- If you’re born, you back and kill your grandfather- Hence you’re born with ½ probability

6. Answer C R CTC R CR 0 0 0 CTC Computation Polynomial Size Circuit “Closed Timelike Curve Register” “Causality-Respecting Register”

7. You (the “user”) pick a circuit C on two registers, RCR and RCTC, as well as an input x to RCR • Let Cx be the induced operation on RCTC only • Nature is forced to find a distribution DCTC over inputs to RCTC such that Cx(DCTC)=DCTC • (If there’s more than one such DCTC, Nature can choose one “adversarially”) • Then Nature samples a string y from DCTC • Output of the computation: C(x,y) PCTCis the class of decision problems solvable in this model

8. How to Use CTCs to Solve Hard Problems: Basic Idea Given a function f:[N]{0,1} (where N is huge), suppose we want “instantly” to find an input x such that f(x)=1 I claim that we can do so using the following function g:[N][N], acting on a CTC register: What are the fixed points of this evolution?

9. mT,0 mT,1 mT-1,0 mT-1,1 m2,0 m2,1 m1,0 m1,1 Theorem:PCTC = PSPACE Proof: For PCTCPSPACE, just need to find some y such that Cx(m)(y)=yfor some m. Pick any y, then apply Cx2n times. For PSPACEPCTC: Have Cxinput and output an ordered pair mi,b, where mi is a state of the Turing machine we’re simulating and b is an answer bit, like so: The only fixed-point distribution is a uniform distribution over all states of the Turing machine, with the answer bit set to its “true” value

10. What About The Quantum Case? • You (the “user”) pick a quantum circuit C on two registers, RCR and RCTC, as well as a (classical) input |x to RCR • Let Cx be the induced superoperator acting on RCTC only • Nature is forced to find a mixed state CTC such that Cx(CTC)=CTC • (If there’s more than one such , Nature can choose one “adversarially”) • Output of the computation: C(x,CTC)

11. Let BQPCTC be the class of problems solvable in this model Certainly PSPACE=PCTCBQPCTCEXP Main Result:BQPCTC = PSPACE “If CTCs are possible, then quantum computers are no more powerful than classical ones”

12. BQPCTCPSPACE: Proof Sketch Let vec() be the “vectorization” of : i.e., a length-22n vector of ’s entries. We can reduce the problem to the following: given an (implicit) 22n22n matrix M, prepare a state CTC in BQPSPACE such that

13. Idea: Let Then • Furthermore: • We can compute P exactly in PSPACE, by using small-space algorithms for matrix inversion discovered in the 1980s (e.g. Csanky’s algorithm) • It’s easy to check that Pv is the vectorization of some density matrix • So then take (say) Pvec(I) as the fixed-point CTC Hence M(Pv)=Pv, so P projects onto the fixed points of M

14. Coping With Error Problem: The set of fixed points could be sensitive to arbitrarily small changes to the superoperator E.g., consider the two stochastic matrices The first has (1,0) as its unique fixed point; the second has (0,1) However, the particular CTC algorithm used to solve PSPACE problems doesn’t share this property! Indeed, one can use a CTC to solve PSPACE problems “fault-tolerantly” (building on Bacon 2003)

15. Discussion • Three ways of interpreting our result: • CTCs exist, so now we know exactly what can be computed in the physical world (PSPACE)! • CTCs don’t exist, and this sort of result helps pinpoint what’s so ridiculous about them • CTCs don’t exist, and we already knew they were ridiculous—but at least we can find fixed points of superoperators in PSPACE! Our result formally justifies the following intuition: By making time “reusable,” CTCs would make time equivalent to space as a computational resource.

16. And Now for the Mudfight! Bennett, Leung, Smith, Smolin 2009: Deutsch’s (and our) model of CTCs is crap Why? Because if you feed to a CTC computer, the outcome might be different than if you fed x and y separately, then averaged the results This is a simple consequence of the fact that CTCs induce nonlinearities in quantum mechanics Bennett et al.’s proposed fix: Force CTC to depend only on the whole distribution over inputs,

17. Our Response What Bennett et al. do basically just amounts to defining CTCs out of existence! Since under their prescription, we might as well treat CTC as a “quantum advice resource” fixed for all time, independent of anything else in the universe That CTCs would strain the normal axioms of physics (like linearity of mixed-state evolution) is obvious … what else did you expect? At least BQPCTC is a good complexity class, better than their proposed replacement BQPPCTC (In any case, our main result—an upper bound on BQPCTC and BQPPCTC—is unaffected)

18. Seth Lloyd’s Response Bennett et al.’s fix precludes the possibility that a CTC could form in some “branches of the multiverse” but not others But quantum gravity theories ought to allow superpositions over different causal structures—so if CTCs can form at all, then why not allow evolutions like

19. Quantum Computing With Closed Timelike Curves BQP Scott Aaronson (MIT) Based on joint work with John Watrous (U. Waterloo) PSPACE